144. Dividing Quantities in a Ratio

Learning Intentions

• To understand that a quantity can be divided into a ratio
• Solve an unknown value for items in a given ratio
• divide a quantity in a particular ratio

Pre-requisite Summary

• Know that a ratio compares quantities
• Be able to write ratios Use the symbol
• Understand that equivalent ratios represent the same comparison
• Be able to Simplify ratios where appropriate
• Know that the total number of parts in a ratio is found by adding the ratio numbers
• Be able to multiply and divide whole numbers accurately
• Understand that one part can be found by dividing the total quantity by the total number of parts

Worked Examples

Worked Example 1

A quantity is divided in the ratio shown. Find the total number of parts:

a)

b)

c)

Worked Example 2

Find the unknown value:

a) If and , find

b) If and , find

Worked Example 3

Find the unknown value:

a) If boys:girls and there are girls, how many boys are there?

b) If red:blue and there are red counters, how many blue counters are there?

Worked Example 4

Divide the quantity in the given ratio:

a) Divide in the ratio

b) Divide in the ratio

Worked Example 5

Divide the quantity in the given ratio:

a) Divide in the ratio

b) Divide $ in the ratio

Worked Example 6

Solve the worded problem:

a) A prize of $ is shared between Amy and Ben in the ratio . How much does each person receive?

b) A rope of length m is cut into two pieces in the ratio . Find the length of each piece.

Problems

Problem 1

A quantity is divided in the ratio shown. Find the total number of parts:

a)

b)

c)

Problem 2

Find the unknown value:

a) If and , find

b) If and , find

Problem 3

Find the unknown value:

a) If boys:girls and there are girls, how many boys are there?

b) If red:blue and there are red counters, how many blue counters are there?

Problem 4

Divide the quantity in the given ratio:

a) Divide in the ratio

b) Divide in the ratio

Problem 5

Divide the quantity in the given ratio:

a) Divide in the ratio

b) Divide $ in the ratio

Problem 6

Solve the worded problem:

a) A prize of $ is shared between Mia and Noah in the ratio . How much does each person receive?

b) A ribbon of length m is cut into two pieces in the ratio . Find the length of each piece.

Exercises

Understanding and Fluency

Exercise 1.

Find the total number of parts in each ratio:

a)

b)

c)

d)

Exercise 2.

Find the unknown value:

a) If and , find

b) If and , find

c) If and , find

Exercise 3.

Find the unknown value:

a) If cats:dogs and there are dogs, how many cats are there?

b) If apples:oranges and there are apples, how many oranges are there?

c) If red:green and there are green counters, how many red counters are there?

Exercise 4.

Divide each quantity in the given ratio:

a) Divide in the ratio

b) Divide in the ratio

c) Divide in the ratio

Exercise 5.

Divide each quantity in the given ratio:

a) Divide in the ratio

b) Divide in the ratio

c) Divide $ in the ratio

Exercise 6.

Solve each ratio problem:

a) A total of stickers is shared in the ratio . How many stickers does each part get?

b) A class has boys:girls in the ratio . If there are boys, how many girls are there?

c) A rope is cut in the ratio and has total length m. Find the two lengths.

Exercise 7.

Write the steps needed to divide a quantity in a ratio:

a) find the total number of parts

b) find the value of one part

c) multiply by each part of the ratio

Exercise 8.

Solve each:

a) Divide in the ratio

b) Divide in the ratio

c) A sum of $ is shared in the ratio . Find each share.

Reasoning

Exercise 9.

Explain why dividing a quantity in the ratio does not mean dividing it into two equal parts.

Exercise 10.

A student says that to divide in the ratio , you should do and . Explain the mistake.

Exercise 11.

Noah says that if the ratio is and one part is worth , then the total is . Is he correct? Explain.

Exercise 12.

Explain why the first step in dividing a quantity in a ratio is to find the total number of parts.

Exercise 13.

A student divides $ in the ratio and writes $ 144,$216$. Describe the error.

Problem-solving

Exercise 14.

A prize of $ is shared between two students in the ratio . Find how much each student receives.

Exercise 15.

A class has boys:girls in the ratio . If there are girls, how many boys are there?

Exercise 16.

A m ribbon is cut into three pieces in the ratio . Find the length of each piece.

Exercise 17.

A farmer divides kg of feed between two animals in the ratio . How much feed does each animal get?

Exercise 18.

A total of $ is shared among three friends in the ratio . Find each share.

Exercise 19.

In a bag, red and blue marbles are in the ratio . If there are blue marbles, how many red marbles are there?

Potential Misunderstandings

• Students may think a ratio such as means the total is always
• Students may divide the quantity by the number of terms in the ratio instead of by the total number of parts
• Students may forget to add all parts of the ratio first
• Students may find the value of one part correctly but then multiply incorrectly
• Students may reverse the order of the ratio when giving answers
• Students may assume the shares are equal because the quantity is being divided
• Students may confuse finding an unknown value from a ratio with simply adding or subtracting the ratio numbers

Next: 145. Scale Drawings and Real Distances